1. Introduction
The suspension system is a critical component of a vehicle, responsible for enhancing both comfort and driving performance. To improve ride quality, the suspension system mitigates road disturbances and minimizes vibrations in the sprung mass. In high-performance vehicles, the suspension system not only suppresses heave and angular motion but also ensures continuous tire contact with the ground. Traditional suspension systems in passenger cars typically employ steel springs and passive dampers. Consequently, once the suspension characteristics are established during the design phase, they cannot be altered during operation. In other words, the behavior of a passive suspension system remains fixed once its target objectives are set [1]. Generally, in passenger cars, the spring and damper settings are optimized for comfort, even at the cost of increased roll and pitch motion. Conversely, sports cars prioritize driving performance by employing stiffer spring and damper settings, even if this compromises ride comfort.
Passive suspension systems are limited in their ability to respond dynamically to changing driving conditions, as their settings are fixed. The development of electronic control suspension (ECS) began in the early 1970s, driven by advancements in electronic control technology. ECS refers to systems that can modify suspension characteristics in real time through electronic control, adapting to various driving conditions. ECS encompasses both semi-active suspension systems, which utilize variable dampers, and active suspension systems, which employ actuators, such as hydraulic servomechanisms or electric motors [2]. Generally, active suspension systems offer superior performance compared to semi-active systems [3]. However, due to the complexities involved in developing actuators for active suspension, semi-active systems using variable dampers were introduced to the market first. When designing a variable damper for semi-active suspension, it is crucial to achieve a broad range of damping force adjustments with minimal energy consumption. In this context, three types of semi-active dampers have been developed: servo/solenoid/piezoelectric valve dampers, magneto-rheological (MR)/electro-rheological (ER) dampers, and electromagnetic dampers [4,5,6,7,8,9,10].
Over the past several decades, various methodologies for the controller design of semi-active suspension systems have been proposed [11]. When assuming the linear behavior of the spring and damper, the suspension model can be represented as a linear time-invariant (LTI) system. Common vehicle models used for suspension analysis include the quarter-car, half-car, and full-car models [12]. Optimal controllers have been developed to calculate the optimal control force, which is then implemented by adjusting the damping force. For instance, an optimal controller based on a quarter-car model has been proposed to minimize suspension deflection, tire deflection, sprung mass acceleration, and jerk [13]. Similarly, an optimal controller can be designed using a half-car model, which accounts for pitch motion and the differences between front and rear suspensions [14]. The linear quadratic (LQ) optimization method has been applied to the linearized full-car model for state vector regulation. In designing the cost function, a comfort index was used to account for heave and pitch accelerations in the frequency domain [15]. Additionally, the target suspension force is determined by a linear quadratic regulator (LQR) within a reciprocal state-space framework [16]. However, the LQR controller based on quarter-car, half-car, and full-car models uses state variables that include physical quantities that are difficult to measure in real vehicles, such as the vertical position of the sprung mass. Therefore, a static output feedback (SOF) control that utilizes only measurable outputs for feedback control was introduced into the suspension controller design [17,18]. In particular, linear quadratic optimization was applied to determine the optimal gain of SOF controllers for suspension [18].
Control strategies based on linear models require robust controller design due to the linearization errors inherent in nonlinear suspension systems. Sliding mode control (SMC) methods have been developed for semi-active suspension systems using linear models. A model reference SMC was initially developed to address modeling errors in the quarter-car model [19], and this approach was later extended to controllers based on the full-car model [20]. To enhance the performance of linear-model-based SMC, parameters of the linear model were identified from the nonlinear model [21]. While SMC improves the robustness of semi-active suspension controllers, dissipativity constraints must be incorporated to account for damper characteristics. In other words, the characteristics of the damper that only dissipates energy must be considered. Model predictive control (MPC) was introduced to explicitly include damper characteristics as constraints in the controller design [22,23,24,25,26]. However, the computational demands of MPC make real-time implementation challenging. To address this, fast MPC algorithms using approximation techniques were developed [24,25], and further advancements led to the exploit MPC, which achieves real-time performance through offline optimization [26].
Model-free approaches have been proposed to directly determine the desired force for semi-active suspension systems. One such approach is PID control, which is used to provide feedback based on the error between the reference signal and the actual outputs. The error in the PID controller can be defined by differences in vertical acceleration [27], vertical velocity [28] of the sprung mass, and the vertical positions of both the sprung and un-sprung masses [29]. Due to the suspension system’s nonlinearity, maintaining a constant PID gain can be challenging under varying operating conditions. To address this, optimization techniques, such as genetic algorithms, have been employed to fine-tune PID controller parameters [27]. Additionally, advanced algorithms, such as the firefly algorithm and particle swarm optimization, have been used to derive optimal gains [28]. More recently, neural networks combined with the sparrow search algorithm have been introduced to determine the optimal PID gains [29].
A new concept in semi-active suspension control, such as skyhook and groundhook control, has been proposed to enhance vehicle dynamics. The skyhook damper theory assumes a virtual damper connected to an inertial reference point in the sky, allowing skyhook control to simultaneously manage resonance and achieve high-frequency isolation [30]. Since skyhook control can be implemented by adjusting the system’s damping coefficient, it is feasible to apply this control strategy in vehicles equipped with variable dampers [31]. Due to these advantages, various skyhook-based semi-active suspension controllers have been developed and refined. For instance, a pre-compensation filter was designed for skyhook controllers to reduce the phase delay typically caused by conventional low-pass filters, improving vibration suppression performance in the 4 Hz to 8 Hz range [32]. Additionally, since actuator delay can impact the effectiveness of semi-active suspensions, an MR damper with reduced control delay was employed to enhance the skyhook control performance [33].
To reduce the number of sensors required for semi-active suspension systems, skyhook control with an acceleration-driven damper was proposed [34]. However, because the acceleration-driven damper is susceptible to high-frequency inputs, a power-driven damper strategy was introduced, combined with the vehicle’s inertial suspension, to mitigate the vertical vibration of the sprung mass. The parameters for skyhook control using a power-driven damper were determined through numerical analysis [35]. An adaptive control approach with soft constraints was also proposed to enhance robustness [36]. To improve skyhook control performance under varying road conditions, a threshold was introduced into the skyhook control law as an operational condition, with its value determined through optimization [37]. Additionally, a hybrid controller was designed to leverage the benefits of both skyhook and groundhook control strategies [38,39]. More recently, a skyhook controller utilizing machine learning has been proposed, which determines the optimal damping coefficient through reinforcement learning [40].
Regardless of the control methods used, various estimators have been developed to estimate vehicle states, suspension parameters, and road profiles. Accurate vehicle state estimation is critical for implementing state feedback control in vehicle platforms. A sliding innovation filter was proposed to estimate the state vector of a quarter-car model, including the position and velocity of the sprung and un-sprung masses, although this approach assumed identical components for both the filter state and measurement vector [41]. A parallel Kalman filter was designed to estimate stroke, stroke rate, velocity of the sprung and un-sprung masses, and damping force; however, this method required a displacement sensor for stroke measurement [42]. Similarly, a Takagi–Sugeno model-based Kalman filter was developed using data from accelerometers and displacement sensors [43]. An adaptive estimation algorithm was introduced to estimate the state of the quarter-car model using two accelerometers for the sprung and un-sprung masses [44]. Additionally, a stroke rate estimator was designed based on a 6D inertial measurement unit (IMU) mounted on the vehicle body, calculating the suspension force [45]. To account for the nonlinearity of the suspension system, a nonlinear parameter-varying observer was proposed for more accurate suspension force estimation [46,47].
For suspension parameter estimation, a model-based estimator was proposed using a radial basis function (RBF) network [48]. The RBF network was employed to approximate the nonlinear mapping of suspension parameters in a linear model. A mass estimator was developed based on a recursive least-square (RLS) algorithm [49]. For the half-car model, the Moore–Penrose pseudo-inverse was introduced to derive suspension parameters through matrix inversion in the frequency domain [50]. Additionally, a controller incorporating parameter estimation was designed to adapt to changes in the mass and inertia of the half-car model [51].
Given that road profiles consist of noise across various frequency bands, estimation has been conducted based on frequency band analysis. A vehicle frequency response function was derived using a Fourier transform of the full-car model to estimate the vehicle frequency response function from the measured data [52,53]. An approximate calculation method was employed to determine the RMS value of power spectral density (PSD) within a time window. To estimate the PSD, the Burg method was used to reduce the computational burden using only a few data records [54]. Similarly, a time-window-based estimator was proposed to estimate the roughness PSD function by using a discrete Fourier transform [55]. Since frequency domain analysis is challenging in vehicles, many studies have focused on indirectly estimating road roughness through suspension state estimation. A Kalman filter with unknown input was used to estimate road elevation [56,57]. To address parameter uncertainty, an interactive, multiple-model adaptive Kalman filter was introduced, enhancing the robustness of state estimation [58]. Recently, machine-learning-based approaches have been developed to directly estimate road roughness from sensor measurements [59,60,61].
A review of previous studies revealed that various estimators and control methods have been proposed for semi-active suspension systems. However, these studies have not adequately addressed scenarios where low-frequency disturbances, such as bumps, coexist with high-frequency disturbances, such as road surface roughness. Since real-world roads present a mix of disturbances across different frequencies, there is a need to develop a semi-active suspension controller capable of effectively responding to both. In other words, this study focused on developing a semi-active suspension control algorithm that can achieve optimal ride comfort for a road surface mixed with low-frequency and high-frequency disturbance.
The passive suspension system cannot actively respond to the driving situation in which low-frequency and high-frequency disturbances are mixed because the determined damping force curve cannot be changed during driving. In addition, for the active suspension system, it is virtually impossible to respond to high-frequency disturbances due to the limitation of the reaction speed of the actuator. Therefore, for the driving situation in which the low-frequency and high-frequency disturbances are mixed, which is to be dealt with in this study, it is required to respond through the damping force control. In particular, since the recent variable damper can adjust the damping force in a wide area, both the high-frequency disturbance response through the damping force curve control and the low-frequency disturbance response through the real-time damping force control can be achieved, thereby exhibiting an effect similar to that of the active suspension. Additionally, it is crucial to develop an estimator that operates with minimal sensors while efficiently estimating suspension conditions and classifying road surface roughness under varying disturbance conditions. To address these challenges, an integrated approach combining adaptive damping control and feedback control is introduced to create a road-adaptive controller for semi-active suspension systems.
The contributions of this paper are as follows:
Integration of the adaptive damping and optimal feedback control. The optimal damping coefficient is adjusted based on road surface roughness and incorporated into the feedback control gain decision, enabling optimal feedback control across varying road conditions.
Minimization of the linear modeling error with linear optimal damping control. The performance of linear-model-based feedback control is maximized by controlling variable dampers to follow linear damping.
Controller design with measurable outputs. The proposed algorithm is designed to utilize only measurable outputs from the actual vehicle, enhancing its practicality and applicability.
Efficient suspension state estimation and road roughness classification. An algorithm is presented that utilizes only a bandwidth filter, the Burg method, and the vehicle’s geometry to efficiently estimate the suspension state and classify road roughness.
The remainder of the paper is organized as follows: Section 2 describes the overall architecture of the road-adaptive static output feedback controller. In Section 3, the vehicle state estimator and road roughness classifier are presented. Section 4 details the design of the feedback and damping control systems. Simulation results, including comparisons with baseline algorithms, are summarized in Section 5. Finally, the conclusion and future work are discussed in Section 6.
5. Simulation Results
The proposed road-adaptive SOF controller for the semi-active suspension system was evaluated through a simulation study. The configuration of the simulation setup is shown in Figure 9. A simulation environment was constructed using a co-simulation of MATLAB/Simulink and CarSim 2017.1. The proposed algorithm is represented as a single block named “semi-active suspension controller”. The vehicle model used was an E-Class Sedan, available as a bundled model in CarSim. The parameters of the vehicle model are summarized in Table 2. Among the parameters of the vehicle model in CarSim, only the parameters related to the half-car model were used for controller design and summarized in Table 2.
Since CarSim provides true measurements of both observable and unobservable states, a sensor model was introduced to produce outputs that closely resemble those of actual IMUs and acceleration sensors. To enhance the agreement between simulation results and real-world vehicle testing, CarSim’s motion sensor function was employed to position the IMU on the sprung mass and acceleration sensors on the left and right lower arms of the front wheels.
For measurements from the IMU, accelerometer, and wheel speed sensors, a sensor model was implemented to mimic the characteristics of real sensors by incorporating sensor delay, amplitude adjustment, and discretization. Additionally, the sampling time of the controller and the system delay of the semi-active damper were aligned with those typically found in vehicle semi-active suspension systems. In this study, the actuator delay was modeled using a first-order time delay model. The parameters for the sensor, controller, and actuator models are summarized in Table 3. Furthermore, the tire–road friction coefficient was set at 0.85, assuming dry asphalt conditions. In addition, the feedback gains for the simulation study are summarized in Table 4.
Among the various vehicle states, vertical acceleration, pitch angle, and pitch rate were used to evaluate the proposed controller. Peak-to-peak (P2P), maximum, and minimum values were selected as key performance indicators (KPIs) for analyzing the simulation results. A total of nine KPIs, along with the time history of vehicle states, were compared between the proposed algorithm and a base algorithm. To assess performance, three comparison cases were established: The first case was a passive setup using the default nonlinear passive damper provided in CarSim. The second case employed only the feedback controller described in Section 4.1 to determine the damping force of the variable damper, with the default damping curve set to the nonlinear curve of the passive damper. This case is named Base #1. The third case was based on a skyhook controller that adapted the damping curve according to road conditions. This case is named Base #2.
A total of four simulation scenarios were designed based on a bump scenario. The bump used in the simulation was 3.6 m long and 10 cm high, in accordance with Korean road construction regulations. The vehicle traveled at 30 kph, using CarSim’s built-in speed controller. The path was set to ensure the vehicle traveled straight over the bump, minimizing roll and lateral motion. Four road roughness profiles were applied to the bump, ranging from an ideal surface with no roughness to profiles corresponding to class A, B, and C roughness levels.
The simulation results for the ideal bump scenario are presented in Figure 10 and Table 5. The vertical acceleration history is shown in Figure 10a. The passive case and Base #2 exhibited higher vertical accelerations compared to the feedback controller. The limited performance improvement of Base #2, despite its damping control, is attributed to the difficulty in responding to low-frequency disturbances solely by adjusting the damping curve based on road roughness. Additionally, the discrete-time estimation of road roughness, necessary for real-time implementation in an embedded environment, prevents precise damping force control at every point on the bump. Conversely, both Base #1 and the proposed controller reduced the peak vertical acceleration. Since the road roughness profile was not applied in the ideal bump scenario, the performance of Base #1 and the proposed controller was nearly identical.
However, different behavior was observed in pitch motion, as depicted in Figure 10b,c. Since Base #2 requires the Burg method’s window duration to update the road class, it reduces pitch motion after the front wheels pass the bump. As a result, the pitch motion is reduced after 2 s with Base #2. Similar to the vertical acceleration, both Base #1 and the proposed controller effectively regulated the pitch angle and pitch rate. Notably, the proposed controller provided better pitch regulation than Base #1, as it adjusted the default damping to behave linearly, satisfying the assumptions of the feedback controller. The KPIs for the ideal bump scenario are summarized in Table 5, where the proposed algorithm demonstrated improvements over the base algorithm in maximum, minimum, and peak-to-peak values for vertical acceleration, pitch angle, and pitch rate. In particular, it can be seen that the improvement in the vertical acceleration was remarkable.
The simulation results for the bump with class A roughness are summarized in Figure 11 and Table 6. When class A roughness was applied, the overall trend mirrored that observed on the ideal road surface. However, the difference between Base #1 and the proposed algorithm became more pronounced due to the adaptation of the default damping. Specifically, the peak values of the proposed algorithm, shown in Figure 11a–c, were further reduced compared to those of Base #1. Additionally, the stroke was more effectively suppressed after the vehicle passed over the bump, as compared to the base algorithms. This overall improvement in performance was further evidenced by the larger differences in KPIs from the base algorithm, as presented in Table 6. Compared to the ideal bump case, where the improvement in the vertical acceleration was large, in the road roughness class A, a clear improvement could be seen not only for vertical acceleration but also for pitch motion.
Figure 12 and Figure 13, along with Table 7 and Table 8, summarize the results for the bump scenario when classes B and C, representing rougher road conditions, were applied. The performance of Base #1, which used only the feedback controller, deteriorated as road roughness increased due to high-frequency disturbances. In contrast, Base #2, which adjusted the default damping according to road roughness, showed a slight improvement in vertical acceleration and pitch behavior. However, because Base #2 struggled to effectively respond to low-frequency disturbances, it did not achieve the same level of performance as Base #1. On the other hand, the proposed algorithm, which addressed both road surface roughness and bump response, maintained ride comfort even under rough road conditions. As shown in Table 7 and Table 8, it can be seen that an improvement in vertical acceleration and pitch motion similar to the results in the ideal bump case appeared even when the road surface was rough.
Source link
Donghyun Kim www.mdpi.com
